Attainable forms of intermediate dimensions

The intermediate dimensions are a family of dimensions which interpolate between the Hausdorff and box dimensions of sets. We prove a necessary and sufficient condition for a given function h(θ) to be realized as the intermediate dimensions of a bounded subset of R ^d . This condition is a straightforward constraint on the Dini derivatives of h(θ), which we prove is sharp using a homogeneous Moran set construction.